mikael27 said:i just know that we are going to have open interval and that the will not be either maximum or minimum. But i don't know how to prove it
mikael27 said:Given that :
(Extreme value theorem). If f : [a; b] in R is continuous, then there exist c, d in [a; b] such that
f(c) = sup{f(x) | x in [a, b]};
f(d) = inf{f(x) | x in [a, b]}
Note that since c, d in [a, b], the supremum and infi mum in the above two equations are in fact the maximum and minimum, respectively.
I tried the follow:
The function f : [0, 1] given by f(x) = 1 for all x in [0, 1] is continuous and for any
c,d in [0; 1] we have
f(c) = 1 = sup{f(x)| x in [0, 1]} = inf{f(x) | x in [0, 1]} = 1 = f(d):
The function f : [0, 1) in R given by f(x) = x is continuous on [0, 1). If
S = {x | x in (0, 1)};
then sup S = 1 and inf S = 0, but these values are not attained. Thus the statement does not hold if [a, b] is replaced by (a, b).
Is this counterexample correct?
mikael27 said:So just say this?
The function f : [0, 1) in R given by f(x) = x is continuous on [0, 1).
then sup(f)=1 ,but there is no point x in [0,1) such that f(x)=1.
Thus the statement does not hold if [a, b] is replaced by (a, b).
The Extreme Value theorem is a mathematical theorem that states that a continuous function on a closed and bounded interval will have both a maximum and minimum value within that interval.
This is because the interval [a; b) is not closed, as it does not include the endpoint b. The Extreme Value theorem requires the interval to be both closed and bounded in order for it to hold.
No, the Extreme Value theorem only applies to closed intervals. An open interval does not have a maximum or minimum value, as the function can continue infinitely in either direction.
If the function is not continuous, the Extreme Value theorem does not apply. The theorem only applies to continuous functions, meaning that there are no gaps or breaks in the graph of the function.
Yes, there are two other conditions that need to be met for the Extreme Value theorem to hold. The first is that the interval must be bounded, meaning that there are finite values for both the upper and lower bounds. The second is that the function must be defined and continuous on the entire interval.